Lesson 5
Reasoning about Equations and Tape Diagrams (Part 2)
5.1: Algebra Talk: Seeing Structure (10 minutes)
Warmup
This warmup parallels the one in the previous lesson. The purpose of this Algebra Talk is to elicit strategies and understandings students have for solving equations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to come up with ways to solve equations of this form. While four equations are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.
Students should understand the meaning of solution to an equation from grade 6 work as well as from work earlier in this unit, but this is a good opportunity to reemphasize the idea.
In this string of equations, each equation has the same solution. Digging into why this is the case requires noticing and using the structure of the equations (MP7). Noticing and using the structure of an equation is an important part of fluency in solving equations.
Launch
Display one equation at a time. Give students 30 seconds of quiet think time for each equation and ask them to give a signal when they have an answer and a strategy. Keep all equations displayed throughout the talk. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Organization
Student Facing
Solve each equation mentally.
\(x 1 = 5 \)
\(2(x1) = 10 \)
\(3(x1) = 15 \)
\(500 = 100(x1)\)
Student Response
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Activity Synthesis
This discussion may go quickly, because students are likely to recognize similarities between this equation string and the one in the previous day’s warmup.
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
 “Who can restate ___’s reasoning in a different way?”
 “Did anyone have the same strategy but would explain it differently?”
 “Did anyone solve the equation in a different way?”
 “Does anyone want to add on to _____’s strategy?”
 “Do you agree or disagree? Why?”
Design Principle(s): Optimize output (for explanation)
5.2: More Situations and Diagrams (15 minutes)
Activity
The purpose of this activity is to work toward showing students that some situations can be represented by an equation of the form \(p(x+q)=r\) (or equivalent). In this activity, students are simply tasked with drawing a tape diagram to represent each situation. In the following activity, they will work with corresponding equations.
For each question, monitor for one student with a correct diagram. Press students to explain what any variables used to label the diagram represent in the situation.
Launch
Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.
Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a wholeclass discussion.
Supports accessibility for: Memory; Language
Student Facing
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
 Each of 5 gift bags contains \(x\) pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
 Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by \(x\) inches so the triangle is still equilateral and has a perimeter of 20 inches.
 An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class.
 Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Ensure students understand that the work of this task is to draw a tape diagram to represent each situation. There is no requirement to write an equation or solve a problem yet.
Arrange students in groups of 2. Give 5–10 minutes to work individually or with their partner, followed by a wholeclass discussion.
Supports accessibility for: Memory; Language
Student Facing
Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.
 Each of 5 gift bags contains \(x\) pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils.
 Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by \(x\) inches so the triangle is still equilateral and has a perimeter of 20 inches.
 An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class.
 Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Select one student for each situation to present their correct diagram. Ensure that students explain the meaning of any variables used to label their diagram. Possible questions for discussion:
 “For the situations with no \(x\), how did you decide what quantity to represent with variable?” (Think about which amount is unknown but has a relationship to one or more other amounts in the story.)
 “What does the variable you used to label the diagram represent in the story?”
 “Did any situations have the same diagrams? How can you tell from the story that the diagrams would be the same?” (Same number of equal parts, same amount for the total.)
Design Principle(s): Optimize output (for explanation); Cultivate conversation
5.3: More Situations, Diagrams, and Equations (10 minutes)
Activity
This activity is a continuation of the previous one. Students match each situation from the previous activity with an equation, solve the equation by any method that makes sense to them, and interpret the meaning of the solution. Students are still using any method that makes sense to them to reason about a solution. In later lessons, a hanger diagram representation will be used to justify more efficient methods for solving.
For each equation, monitor for a student using their diagram to reason about the solution and a student using the structure of the equation to reason about the solution.
Launch
Keep students in the same groups. 5 minutes to work individually or with a partner, followed by a wholeclass discussion.
Supports accessibility for: Language; Socialemotional skills
Design Principle(s): Support sensemaking; Maximize metaawareness
Student Facing
Each situation in the previous activity is represented by one of the equations.
 \((x+3) \boldcdot 5 = 20\)
 \(3(x+5)=20\)
 Match each situation to an equation.
 Find the solution to each equation. Use your diagrams to help you reason.

What does each solution tell you about its situation?
Student Response
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Student Facing
Are you ready for more?
Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting?
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
For each equation, ask one student who reasoned with the diagram and one who reasoned only about the equation to explain their solutions. Display the diagram and the equation side by side, drawing connections between the two representations. If no students bring up one or both of these approaches, demonstrate the maneuvers on a diagram side by side with the maneuvers on the corresponding equation. For example, “I divided the number of gift bags by 5, leaving me with 4 pencils per gift bag. Since Tyler added 3 pencils to each gift bag, there must have been 1 pencil in each gift bag to start,” can be shown on a tape diagram and on a corresponding equation. It is not necessary to invoke the more abstract language of “doing the same thing to each side” of an equation yet.
Lesson Synthesis
Lesson Synthesis
Display one of the situations from the lesson and its corresponding equation. Ask students to explain:
 “What does each number and letter in the equation represent in the situation?”
 “What is the reason for each operation (multiplication or addition) used in the equation?”
 “What is the solution to the equation? What does it mean to be a solution to an equation? What does the solution represent in the situation?”
5.4: Cooldown  More Finding Solutions (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Equations with parentheses can represent a variety of situations.
 Lin volunteers at a hospital and is preparing toy baskets for children who are patients. She adds 2 items to each basket, after which the supervisor’s list shows that 140 toys have been packed into a group of 10 baskets. Lin wants to know how many toys were in each basket before she added the items.
 A large store has the same number of workers on each of 2 teams to handle different shifts. They decide to add 10 workers to each team, bringing the total number of workers to 140. An executive at the company that runs this chain of stores wants to know how many employees were in each team before the increase.
Each bag in the first story has an unknown number of toys, \(x\), that is increased by 2. Then ten groups of \(x+2\) give a total of 140 toys. An equation representing this situation is \(10(x+2)=140\). Since 10 times a number is 140, that number is 14, which is the total number of items in each bag. Before Lin added the 2 items there were \(14  2\) or 12 toys in each bag.
The executive in the second story knows that the size of each team of \(y\) employees has been increased by 10. There are now 2 teams of \(y+10\) each. An equation representing this situation is \(2(y+10)=140\). Since 2 times an amount is 140, that amount is 70, which is the new size of each team. The value of \(y\) is \(7010\) or 60. There were 60 employees on each team before the increase.